“God created the integers and the rest is the work of man.” (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886). In this chapter, we go over the essential and foundational facts about the real number system. First, we will provide the structure of real numbers through certain axioms, and then we will show that between any two real numbers, there is an infinite number of rational and irrational numbers, which in turn are used to prove several interesting results.